Step-by-Step Guide to Finding Increasing and Decreasing Intervals
The concept of increasing and decreasing intervals is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding these intervals allows students to analyze and predict the behavior of functions, especially in calculus and algebra, and is crucial for board and competitive exams.
Understanding Increasing and Decreasing Intervals
An increasing and decreasing interval describes the specific ranges on the x-axis where a function is either rising (increasing) or falling (decreasing) as you move from left to right. This concept is widely used in calculus, monotonicity, and while solving application of derivatives problems. It is also important in graph analysis and helps students recognize turning points on a curve.
Formula Used in Increasing and Decreasing Intervals
The standard formula to determine these intervals is by checking the sign of the first derivative of the function:
If \( f'(x) > 0 \) on an interval, the function is increasing there.
If \( f'(x) < 0 \) on an interval, the function is decreasing there.
Here’s a helpful table to understand increasing and decreasing intervals more clearly:
Increasing and Decreasing Intervals Table
| Interval | Sample x | Derivative f'(x) | Type |
|---|---|---|---|
| (-∞, -5) | x = -6 | > 0 | Increasing |
| (-5, 3) | x = 0 | < 0 | Decreasing |
| (3, ∞) | x = 4 | > 0 | Increasing |
This table shows how the sign of the derivative classifies intervals as increasing or decreasing. You can use similar steps for quadratic and polynomial functions.
How to Find Increasing and Decreasing Intervals
Follow these steps for any function:
1. Differentiate the function to get \( f'(x) \).
2. Set \( f'(x) = 0 \) and solve for x. These are the critical points.
3. Mark out intervals between and beyond the critical points (for example, \((-\infty,a), (a,b), (b,\infty)\)).
4. Pick a test value from each interval and substitute into \( f'(x) \) to check if it is positive or negative.
5. If \( f'(x) > 0 \) in an interval, the function is increasing there. If \( f'(x) < 0 \), it is decreasing.
Worked Example – Solving a Problem
Let’s find the increasing and decreasing intervals for \( f(x) = x^3 + 3x^2 - 45x + 9 \).
1. Differentiate: \( f'(x) = 3x^2 + 6x - 45 \)
2. Factor the derivative: \( f'(x) = 3(x + 5)(x - 3) \)
3. Set \( f'(x) = 0 \): \( x = -5 \), \( x = 3 \)
4. Test intervals:
For \( -5 < x < 3 \), choose \( x = 0 \): \( f'(0) = -45 < 0 \) → Decreasing
For \( x > 3 \), choose \( x = 4 \): \( f'(4) = 27 > 0 \) → Increasing
5. Final Answer:
The function is increasing on \( (-\infty, -5) \) and \( (3, \infty) \), and decreasing on \( (-5, 3) \).
Practice Problems
- Find the increasing and decreasing intervals of \( f(x) = 2x^2 - 8x + 1 \).
- Determine whether \( f(x) = -x^3 + 3x^2 + 9 \) is increasing or decreasing between x = -1 and x = 4.
- For \( f(x) = x^4 \), list all intervals where the function is increasing.
- Check if \( f(x) = 3x + 5 \) is always increasing, always decreasing, or neither.
Common Mistakes to Avoid
- Confusing increasing and decreasing intervals with where the function value is simply high or low.
- Forgetting to check open and closed intervals while writing interval notation.
- Assuming a function is always increasing or always decreasing without examining the derivative.
- Not substituting test points from each interval into the derivative.
Real-World Applications
The concept of increasing and decreasing intervals is used to study trends in finance, biology (population models), physics (particle motion), and economics (profit functions). With graphing, you can spot where values rise or fall, helping to make wise decisions. Vedantu helps students link these ideas to both board exams and real-life scenarios.
Page Summary
We explored the idea of increasing and decreasing intervals, learned how to find them using derivatives, practiced with examples, and understood how to avoid common mistakes. Practice more problems with Vedantu to become confident in identifying intervals and applying these skills in all exams.
Further Reading
For a deeper understanding of increasing and decreasing intervals, you can also review these topics:
Increasing and Decreasing Functions and Monotonicity |
Derivatives |
Application of Derivatives |
Maxima and Minima Using First Derivative Test
FAQs on How to Determine If a Function Is Increasing or Decreasing
1. How do you check if a function is increasing or decreasing on a given interval?
To determine if a function is increasing or decreasing on an interval, you analyze the sign of its derivative (f'(x)) within that interval. If f'(x) > 0 for all x in the interval, the function is increasing. If f'(x) < 0, the function is decreasing. This method is commonly used in calculus and applies to polynomials, quadratics, and most continuous functions.
2. Is f(x) = 4x + 3 increasing or decreasing?
The function f(x) = 4x + 3 is strictly increasing on its entire domain. This is because its derivative is f'(x) = 4, which is always positive. Thus, as x increases, so does f(x).
3. How do you find the increasing and decreasing intervals of a polynomial?
To find the increasing and decreasing intervals of a polynomial:
- Find the derivative f'(x) of the polynomial.
- Set f'(x) = 0 to identify critical points.
- Test intervals between critical points by plugging values into f'(x):
- If f'(x) > 0 in an interval, the function is increasing there.
- If f'(x) < 0, the function is decreasing.
4. How can you identify strictly increasing and strictly decreasing intervals?
An interval is strictly increasing if the derivative f'(x) > 0 at every point in that interval. It is strictly decreasing if f'(x) < 0 throughout the interval. There should be no points in the interval where the derivative is zero.
5. How do you find increasing and decreasing intervals on a graph?
On a graph, an increasing interval appears as a section where the curve moves upward from left to right, while a decreasing interval moves downward. Look for where the slope is positive (rises up) for increasing, and negative (falls down) for decreasing.
6. What are the increasing and decreasing intervals of a quadratic function?
For a standard quadratic function f(x) = ax^2 + bx + c:
- If a > 0, the function decreases until the vertex, then increases.
- If a < 0, it increases until the vertex, then decreases.
- The vertex divides the function into its increasing and decreasing intervals.
7. How do you use derivatives to find increasing and decreasing intervals?
You use the first derivative test by finding where f'(x) = 0 (critical points) and checking the sign of f'(x) on intervals around those points. Where f'(x) > 0, the function is increasing; where f'(x) < 0, it is decreasing.
8. What is the definition of decreasing interval?
A decreasing interval is a portion of the domain of a function where as x increases, the value of the function f(x) decreases. Mathematically, for any two numbers x₁ < x₂ in the interval, f(x₁) > f(x₂).
9. Can you use a calculator to find increasing and decreasing intervals?
Yes, many graphing calculators and online tools can help locate increasing and decreasing intervals by plotting the function and its derivative, making it easier to identify intervals where the function rises or falls.
10. Give an example showing increasing and decreasing intervals of a polynomial.
For f(x) = x^3 - 3x^2:
- f'(x) = 3x^2 - 6x
- Set f'(x) = 0: 3x(x-2)=0 ⇒ x=0, x=2
- Test intervals:
- x < 0: f'(x) > 0 (increasing)
- 0 < x < 2: f'(x) < 0 (decreasing)
- x > 2: f'(x) > 0 (increasing)
11. How do you write increasing and decreasing intervals?
You write increasing or decreasing intervals using interval notation. For example, a function increasing on (-∞, 1) and (3, ∞) would be written as 'Increasing on (-∞, 1) ∪ (3, ∞)'.
12. What do increasing and decreasing intervals indicate about a function's graph?
The increasing and decreasing intervals show where the function is going upwards or downwards on its graph. These intervals often identify local maxima and minima, or turning points, which are important for understanding the function's shape and behavior.
